Optimal. Leaf size=70 \[ \frac{i a \text{Chi}\left (x f+\frac{c f}{d}\right ) \sinh \left (e-\frac{c f}{d}\right )}{d}+\frac{i a \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d}+\frac{a \log (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.161418, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3317, 3303, 3298, 3301} \[ \frac{i a \text{Chi}\left (x f+\frac{c f}{d}\right ) \sinh \left (e-\frac{c f}{d}\right )}{d}+\frac{i a \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d}+\frac{a \log (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3317
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{a+i a \sinh (e+f x)}{c+d x} \, dx &=\int \left (\frac{a}{c+d x}+\frac{i a \sinh (e+f x)}{c+d x}\right ) \, dx\\ &=\frac{a \log (c+d x)}{d}+(i a) \int \frac{\sinh (e+f x)}{c+d x} \, dx\\ &=\frac{a \log (c+d x)}{d}+\left (i a \cosh \left (e-\frac{c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{c f}{d}+f x\right )}{c+d x} \, dx+\left (i a \sinh \left (e-\frac{c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{c f}{d}+f x\right )}{c+d x} \, dx\\ &=\frac{a \log (c+d x)}{d}+\frac{i a \text{Chi}\left (\frac{c f}{d}+f x\right ) \sinh \left (e-\frac{c f}{d}\right )}{d}+\frac{i a \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (\frac{c f}{d}+f x\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.29515, size = 60, normalized size = 0.86 \[ \frac{a \left (i \text{Chi}\left (f \left (\frac{c}{d}+x\right )\right ) \sinh \left (e-\frac{c f}{d}\right )+i \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (f \left (\frac{c}{d}+x\right )\right )+\log (c+d x)\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.045, size = 96, normalized size = 1.4 \begin{align*}{\frac{a\ln \left ( dx+c \right ) }{d}}+{\frac{{\frac{i}{2}}a}{d}{{\rm e}^{{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,fx+e+{\frac{cf-de}{d}} \right ) }-{\frac{{\frac{i}{2}}a}{d}{{\rm e}^{-{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,-fx-e-{\frac{cf-de}{d}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.20787, size = 96, normalized size = 1.37 \begin{align*} \frac{1}{2} i \, a{\left (\frac{e^{\left (-e + \frac{c f}{d}\right )} E_{1}\left (\frac{{\left (d x + c\right )} f}{d}\right )}{d} - \frac{e^{\left (e - \frac{c f}{d}\right )} E_{1}\left (-\frac{{\left (d x + c\right )} f}{d}\right )}{d}\right )} + \frac{a \log \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.77263, size = 162, normalized size = 2.31 \begin{align*} \frac{i \, a{\rm Ei}\left (\frac{d f x + c f}{d}\right ) e^{\left (\frac{d e - c f}{d}\right )} - i \, a{\rm Ei}\left (-\frac{d f x + c f}{d}\right ) e^{\left (-\frac{d e - c f}{d}\right )} + 2 \, a \log \left (\frac{d x + c}{d}\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int \frac{i \sinh{\left (e + f x \right )}}{c + d x}\, dx + \int \frac{1}{c + d x}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.29067, size = 96, normalized size = 1.37 \begin{align*} -\frac{i \, a{\rm Ei}\left (-\frac{d f x + c f}{d}\right ) e^{\left (\frac{c f}{d} - e\right )} - i \, a{\rm Ei}\left (\frac{d f x + c f}{d}\right ) e^{\left (-\frac{c f}{d} + e\right )} - 2 \, a \log \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]